An identity about floor function.

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An identity about floor function.

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#1 h Jun 7, 2025, 2:56 AM

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How to prove this identity in wikipedia?It's about floor function.
https://en.wikipedia.org/wiki/Floor_and_ceiling_functions#Quotients

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This post has been edited 1 time. Last edited by Tony_Yin_Math, Jun 7, 2025, 2:57 AM
Reason: Add an link of wikipedia

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#2 h Jun 7, 2025, 8:03 AM • 1 Y

Y by Tony_Yin_Math

This is basically counting lattice points in a rectangle. Consider the rectangle $(0,0),(n,0),(n,m),(0,m)$ and the diagonal $y={{mx}\over n}$ .
- total number of points is $(m+1)(n+1)$ .
- total number on the diagonal is $d+1$ where $d:=(m,n)$ , the gcd.
Let $A$ be the set of lattice points in the interior of the rectangle, under (not on) the diagonal.
Then $A=\sum_1^{n-1}[km/n]-(d-1)$ . Also $2|A|+2(m+n)+(d-1)=(m+1)(n+1)$ .
Combining gives the required formula. Welcome to AOPS.

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#3 h Jun 7, 2025, 8:07 AM

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Thank you very much! I ask Gemini, and get the very same answer. Before I ask Gemini, I ask Deepseek R1, and Deepseek R1 can't give the answer.

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